Forced oscillations · Differential Equations

Intuition

Push a damped spring with a force that swings back and forth like a cosine, and it ends up moving at the frequency of the push, not its own. Every solution is the particular solution plus a homogeneous one, and with friction the homogeneous part — the transient — dies away. What remains is the steady state, a cosine at the driving frequency, whose amplitude depends on how the driving frequency compares with the natural one.

A child on a swing pushed in a steady rhythm. Whatever the swing was doing when the pushing began is soon forgotten, and it ends up swinging in time with the pushes.

$x''+0.4x'+x=\cos1.5t$ from two different starts. The transients differ and die away, and both motions settle onto the same steady state, at the driving frequency $1.5$ .

Transient and steady state

For $mx''+cx'+kx=F_{0}\cos\omega t$ with , the particular solution is with the amplitude below. Every solution is plus a homogeneous solution, which decays; so after the transient every motion is , whatever the initial conditions.

A(\omega)=\frac{F_{0}}{\sqrt{(k-m\omega^{2})^{2}+(c\omega)^{2}}}

Reading the response

The steady state has the driving frequency $\omega$ , not the natural one.
The initial conditions affect only the transient, and the transient dies away.
The amplitude is largest when $\omega$ is near $\omega_{0}=\sqrt{k/m}$ and the damping is small; far from it the response is small.

Every forced motion approaches the same steady state

c>0\ \Longrightarrow\ x(t)-x_{p}(t)\to0\ \text{for every solution}\ x

The difference between any solution and the particular solution solves the homogeneous equation, by linearity. With positive damping every homogeneous solution dies away, as the damping lesson proved. So the difference tends to zero: every motion, whatever its start, approaches the one particular motion.

Proof steps

m(x-x_{p})''+c(x-x_{p})'+k(x-x_{p})=0

The difference of two solutions solves the homogeneous equation.

c>0\ \Longrightarrow\ x-x_{p}\to0

With damping, every homogeneous solution dies away.

x(t)\approx x_{p}(t)\ \text{for large}\ t

So the start is forgotten, and only the steady state remains.

Applications

Practice

It Moves at the Driving Frequency

After the transient, a damped system driven by a cosine moves as a cosine of the same frequency, with its own amplitude and phase.

x_{p}=A\cos(\omega t-\phi)

Try it

A damped spring with natural frequency $3$ is driven by $\cos5t$ . At what frequency does it move in the long run?